Use the appropriate addition formula to find the exact value of the expression.
-$\cos \left( - \frac { 7 \pi } { 12 } \right)$
A) $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$
B) $\frac { \sqrt { 2 } - \sqrt { 6 } } { 4 }$
C) $\sqrt { 2 } - \sqrt { 6 }$
D) $\sqrt { 6 } + \sqrt { 2 }$

Question

Quizplus · Accepted Answer

Starting with the formula for $\cos(a-b)$, we have $\cos\left(-\frac{7\pi}{12}ight)=\cos\left(\frac{\pi}{4}-\frac{5\pi}{12}ight)$. We can then use the addition formula for cosine: $$\cos(a-b)=\cos a\cos b+\sin a\sin b$$ Setting $a=\frac{\pi}{4}$ and $b=\frac{5\pi}{12}$, we get:
\begin{align*}
\cos\left(\frac{\pi}{4}-\frac{5\pi}{12}ight)&=\cos\frac{\pi}{4}\cos\frac{5\pi}{12}+\sin\frac{\pi}{4}\sin\frac{5\pi}{12}\
&=\frac{\sqrt2}{2}\cdot\frac{\sqrt{6}-\sqrt2}{4}+\frac{\sqrt2}{2}\cdot\frac{\sqrt{6}+\sqrt2}{4}\
&=\frac{\sqrt2\sqrt{6}}{8}\
&=\boxed{\frac{\sqrt2-\sqrt6}{4}}\
\end{align*}

Use the Appropriate Addition Formula to Find the Exact Value cos⁡(−7π12)\cos \left( - \frac { 7 \pi } { 12 } \right)cos(−127π​)

Use the Appropriate Addition Formula to Find the Exact Value $\cos \left( - \frac { 7 \pi } { 12 } \right)$